[Math] Getting the story of Dynkin and Satake diagrams straight

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I've been trying to teach myself the theory of Lie groups. The sources I've been reading reference Lie algebras in the context of Dynkin and Satake diagrams, but not Lie groups (which I am more interested in). I've been trying to translate these results to Lie groups, but I'm almost sure I'm getting my facts wrong. (I am in particular confused about what is true over $\mathbb{C}$, and what is true over $\mathbb{R}$ because some sources don't mention this explicitly.)

It would be very helpful for me to get some guidance so that I know how to think about the big story before continuing with my study.

Question

Is the following story correct? (And if not, what is the correct story?)

  1. Semi-simple real Lie algebras correspond to disjoint unions of Satake diagrams.

  2. Semi-simple real Lie algebras are always the Lie algebra of some real compact Lie group. This Lie group is not unique, but it is unique if we assume that it is connected and simply connected.

  3. Any connected compact real Lie group with a semi-simple (real) Lie algebra is semi-simple. Namely, it is reductive with finite center.

  4. Any compact real Lie group is a maximal compact closed subgroup of its complexification. (Although there may be other maximal compact subgroups.)

  5. A complex Lie group is reductive iff it is the complexification of a compact real Lie group.

  6. The maximal closed compact subgroups of a complex reductive Lie group are all real Lie groups.

  7. The maximal closed compact subgroups of a complex reductive Lie group correspond to disjoint unions of Dynkin diagrams.

Semi-simple complex Lie algebras correspond to disjoint unions of Dynkin diagrams.

$\,\,\,$8. A semi-simple complex Lie algebra is always the Lie algebra of a reductive complex Lie group. (This group may not be unique. I am not sure whether it is unique if we assume that it's simply connected as a manifold.)

EDIT: the point below did not appear in the original question

$\,\,\,$9. Non-isomorphic real (resp. complex) connected and simply connected Lie groups have non-isomorphic real (resp. complex) Lie algebras.

Additional Question

In addition to my confusion about the story above, I am also confused about the way in which this helps classify real Lie groups in general. Namely, if I understand correctly, Satake diagrams only classify connected compact simply connected real Lie groups; and the connected compact real Lie groups are known because they are quotients of connected compact simply connected real Lie groups by central subgroups. But what about non-connected compact real Lie groups? Are they completely unrelated to the Satake diagrams story? Is there any way to classify them?

EDIT: The following is another point of confusion that did not appear in the original question

Let's say we are given a semi-simple complex Lie algebra, and let's say that it's the Lie algebra of some reductive connected and simply connected complex Lie group. Since Dynkin diagrams are a coarser tool than Satake diagrams, I take from that the this complex Lie algebra is potentially the complexification of several non-isomorphic real Lie algebras. However, if I understand correctly, a complex Lie group is the complexification of at most one (up to iso.) real Lie group. So how could it be that the complex Lie algebra has several real forms, but the complex Lie group has only one? What are these real Lie groups that correspond to the various real forms of the complex Lie algebra?

Never mind, I got it. Satake diagrams correspond to not-nec.-compact, connected, simply connected, real Lie groups.

Best Answer

2 is false. The smallest counterexample is $\mathfrak{sl}_2(\mathbb{R})$. A necessary and sufficient condition for a semisimple real Lie algebra to be the Lie algebra of a compact Lie group is that the Killing form is negative definite (compact Lie algebra). I think it is known that every semisimple complex Lie algebra has a unique compact real form.

But what about non-connected compact real Lie groups? Are they completely unrelated to the Satake diagrams story? Is there any way to classify them?

This is at least as hard as classifying finite groups.

Edit: Regarding 9, a much stronger statement is true. Taking tangent spaces at the identity induces an equivalence of categories between the category of connected, simply connected real Lie groups and the category of finite-dimensional real Lie algebras. I think this statement is still true with "real" replaced by "complex."

Never mind, I got it. Satake diagrams correspond to not-nec.-compact, connected, simply connected, real Lie groups.

By the above, classifying these is equivalent to classifying finite-dimensional real Lie algebras, and this classification is hopeless already for nilpotent Lie algebras of some specific small dimension that I can't remember right now. Statement 1 is still correct.

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